Find the derivative of the function.y=(8x3−2x)3/2y=(8x^3-2x)^{3/2}y=(8x3−2x)3/2

( 36 x 2 − 3 ) 8 x 3 − 2 x \left(36x^2-3\right)\sqrt{8x^3-2x} ( 36 x 2 − 3 ) 8 x 3 − 2 x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Find the derivative of the function. y = ( 8 x 3 − 2 x ) 3 / 2 y=(8x^3-2x)^{3/2} y = ( 8 x 3 − 2 x ) 3/2

this problem, we're asked to find the derivative of the function y equals 8x cube minus 2x. All of that is raised to the power of 3 halves. Now since here I see that I have one function inside of another function, I know I need to apply the chain rule, working my way from the outside in. Now here I can see that my inside function is that 8x cubed minus 2x, making my outside function, all of that raised to the power of 3 halves. Now working our way from the outside in, we, of course, wanna start by working with our outside function. So in finding our derivative here, y prime, starting with that outside function, I wanna use the power rule, taking that 3 halves, my power, and multiplying by it. So I have 3 halves in the front here. Then I am just keeping all of that stuff inside the parentheses the same because, remember, we're treating that as though it's basically a variable. We're only taking the derivative of our outside function now. Now I wanna decrease that power by 1. 3 halves minus 1 because that becomes 3 halves minus 2 halves because that's 1. This is going to give me 1 half. But we are not done yet. Of course, we're using the chain rule. We're working our way inside. So now we want to take the derivative of our inside function multiplying by it. Now here I'm going to go ahead and use the power rule as usual. So taking the power rule here, I'm multiplying by that 3. 3 times 8 is going to give me 24 and then decreasing that power by 1. That becomes x to the power of 2. Then I am subtracting here. So a minus 2x, the derivative of this term. Since I have a constant multiplying x, I know that my answer is just my constant. So I have a minus 2. Now we can rewrite this in a bit more compact way and we can also make this one half into a square root. Remember that this is still a correct answer but we always want to try to simplify as much as we can. And remember that sometimes your answer might not look exactly like mine, but it still might be an equivalent statement. So make sure and double check that there are not other ways that you could have simplified, and you still may have a right answer. So let's go ahead and simplify this a little bit more. Now I'm gonna go ahead and put all of that in the square root, and I am also going to take this 3 halves and distribute it through these terms just so that I don't have to worry about that being pulled out. So if I go ahead and multiply 24x squared and negative 2 by 3 halves, this is going to end up giving me 36x squared minus 3. So that's that first term. So I've accounted for this and I have accounted for this. Now all of this is being multiplied by that square root of 8x cubed minus 2x. All of that is under that square root. This is my derivative here simplified as much as I can. Y prime is equal to 36x squared minus 3 times the square root of 8x cubed minus 2x. Now you may want to do some factoring here because I see I have this 36 and this negative 3, but it's not absolutely necessary unless that is told explicitly to you to do by your instructor. Feel free to let me know if you have any questions here, and we're gonna get some more practice coming up. So I'll see you there.

Find the derivative of the function. y = ( 8 x 3 − 2 x ) 3 / 2 y=(8x^3-2x)^{3/2} y = ( 8 x 3 − 2 x ) 3/2

( 36 x 2 − 3 ) 8 x 3 − 2 x \left(36x^2-3\right)\sqrt{8x^3-2x} ( 36 x 2 − 3 ) 8 x 3 − 2 x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

3 2 24 x 2 − 2 \frac32\sqrt{24x^2-2} 2 3 24 x 2 − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

3 2 24 x 2 − 2 \frac{3}{2\sqrt{24x^2-2}} 2 24 x 2 − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> 3

( 12 x 2 − 1 ) 8 x 3 − 2 x \left(12x^2-1\right)\sqrt{8x^3-2x} ( 12 x 2 − 1 ) 8 x 3 − 2 x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

3. Techniques of Differentiation

The Chain Rule

Business Calculus

3. Techniques of Differentiation

The Chain Rule

Struggling with Business Calculus?

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Multiple Choice

Find the derivative of the function.
$y=(8x^3-2x)^{3/2}$

$\frac32\sqrt{24x^2-2}$

$\left(36x^2-3\right)\sqrt{8x^3-2x}$

$\frac{3}{2\sqrt{24x^2-2}}$

$\left(12x^2-1\right)\sqrt{8x^3-2x}$

Verified step by step guidance

Step 1: Recognize that the function y = (8x^3 - 2x)^(3/2) is a composite function. To differentiate it, we will use the chain rule. The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Step 2: Let the inner function be u = 8x^3 - 2x. Then the outer function becomes y = u^(3/2). Differentiate the outer function with respect to u: dy/du = (3/2) * u^(1/2).

Step 3: Differentiate the inner function u = 8x^3 - 2x with respect to x: du/dx = 24x^2 - 2.

Step 4: Combine the results from Step 2 and Step 3 using the chain rule: dy/dx = (dy/du) * (du/dx). Substituting, we get dy/dx = (3/2) * (8x^3 - 2x)^(1/2) * (24x^2 - 2).

Step 5: Simplify the expression by combining terms. The final derivative is dy/dx = (36x^2 - 3) * sqrt(8x^3 - 2x).

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